Common to pool data or consider problems on a region by region basis.


This can make statistical problems more tractable.

National Resource Management Regions

Whan, Kirien, and Maurice Schmeits. "Comparing area probability forecasts of (extreme) local precipitation using parametric and machine learning statistical postprocessing methods." Monthly Weather Review 146.11 (2018): 3651-3673.

How should assign regions for the analysis of extremes?

Create regions that are likely to experience similar impacts

1. Regionalisation

• Clustering
• Dependence of bivariate extremes
• Practicalities
• Classification

2. Visualise spatial dependence

• Max-stable processes

3. Spatial post-processing

Require: Notion of closeness between two locations


Want: Form clusters based on extremal dependence


Solution: The F-madogram distance

Bernard, Elsa, et al. "Clustering of maxima: Spatial dependencies among heavy rainfall in France." Journal of Climate 26.20 (2013): 7929-7937.

\[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).

Advantages:

• Only use the raw block (annual) maxima
• No information about climate or topography
• Non-parametric estimation (fast)

Cooley, D., Naveau, P. and Poncet, P., 2006. Variograms for spatial max-stable random fields. In Dependence in probability and statistics (pp. 373-390). Springer, New York, NY.

For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals, \(\theta(x_i - x_j)\) is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\]

The range of \(\theta(x_i - x_j)\) is \([1 , 2]\).

Can write our distance measure as a function of the extremal coefficient, \(\theta(x_i - x_j)\), \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)}.\]

Therefore the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).

1. Randomly select an initial set of \(K\) stations. These are the set of the initial medoids.
2. Assign each station, \(x_i\), to its closest medoid, \(m_k\), based on the F-madogram distance.
3. For each cluster, \(C_k\), update the medoid according to \[m_k = \mathop{\mathrm{argmin}}\limits_{x_i \in C_k} \sum_{x_j \in C_k} d(x_i, x_j).\]
4. Repeat steps 2. – 4. until the medoids are no longer updated.

Kaufman, L. and Rousseeuw, P.J., 1990. Partitioning around medoids (PAM). Finding groups in data: an introduction to cluster analysis, pp.68-125.

Consider the \(\max \{ \| x_i - x_j \|, 2\}\) as the clustering distance.

Spatial density is changed by land-sea and domain boundaries

Tendancy toward clusters of equal size

Clustering is in F-madogram space not Euclidean

1. Each station starts in its own cluster
2. For each pair of clusters, \(C_k\) and \(C_k'\), define the distance between the clusters as \[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]
3. Merge the the clusters with the smallest distance
4. Update the distances relative to the new cluster
5. Repaet steps 3 - 5, until all points are combined in a single cluster

• Classify a station relative to its closest neighbours
• Use a weighted classification \(w\)-kNN

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Where can we assume a common dependence structure for extremes?

Shiny App

SWWA

TAS

Oesting et. al 2017

• approach

• cut the region into two